Marywood University Department of Mathematics Course Information for Math 311 by Dr. Craig Johnson

Math 311 Differential Equations

Trajectories in the phase plane of a 2x2 first-order system. The eigenvalues are real and of opposite sign.

The study of ordinary differential equations and first-order systems (linear and nonlinear) through a combination of analytical, numerical, and qualitative techniques. Topics include direction fields, solving first order equations of several types, solving second order linear equations, the Laplace transform, first order systems, phase plane portraits, and classification of equilibrium points. The textbook is "Elementary Differential Equations" by Boyce/DiPrima and published by Wiley. The course grade is determined by 3 tests, 4 homework sets, 2 Maple computer labs, a semester project, and the final examination. The project must demonstrate a substantial application of differential equations. Related articles from the American Mathematical Monthly and useful websites for the course are listed at the end of this page. Prerequisite: Calculus II

Section	Topic	Section	Topic	Section	Topic
1.1	Direction Fields	3.1	Homogeneous Eqns	6.1	Laplace Transform
1.2	Some solutions of DE's	3.2	Fundamental Solns	6.2	Solution of IVP's
1.3	Classification of ODE's	3.3	Linear Independence	6.6	Convolution Integral
2.1	Linear Equations	3.4	Complex Roots	7.2	Review of Matrices
2.2	Separable Equations	3.5	Repeated Roots	7.3	Eigenvectors
2.3	Modeling	3.6	Undetermined Coeffts	7.4	Systems of 1st order de
2.4	Differences of Lin vs Nonlin	3.7	Variation of Parameters	7.5	Homog. Linear Systems
2.5	Autonomous Eqns	3.8	Mech & Electrical Vibrations	7.6	Complex Eigenvalues
2.6	Exact Eqns	4.1	Theory of nth order eqns	7.8	Repeated Eigenvalues
2.7	Euler's Method	4.2	Homog. nth order eqns	9.1	The Phase Plane
2.8	Existence and Uniqueness			9.2	Autonomous Systems
	TEST #1		TEST #2		TEST #3

Sample Problems for Test #1

• Consider the d.e. (y''')² + 32y⁵y' = 0.
  • i) Is this equation linear or nonlinear?
  • ii) What is the order of the equation?
  • iii) Is y = e^-2x a solution? Justify your answer.
• Solve the linear d.e. (cos²x sin x) dy + (y cos³x - 1) dx = 0.
• Solve: dy/dx = (y² + 2xy)/x²
• Solve the IVP: dy/dx = (8x + 6xy³ + 1) / (3 - 9x²y² - 3y² ) , y(0) = 4
•  
• A retired person has a sum S(t) invested so as to draw interest at an annual rate r compounded continuously. Withdrawals for living expenses are made (continuously) at a rate of k dollars per year.
  • i) If the initial value of the investment is So dollars, set up the differential equation and solve it to determine S(t) at any time.
  • ii) Assuming that So and r are fixed, determine the withdrawal rate k at which S(t) will remain constant.
•  
• On his last case, Inspector I. "Sherlock" Newton discovered a dead body in a restaurant freezer (temp 28° F) and immediately took the temperature of the body. He found it to be 60° F. Thirty minutes later the body's temp had dropped to 45° F. After years of brilliant detective work Sherlock had formulated his famous law of cooling which states: The time rate of change of the temperature q = q(t) of a body at time t is proportional to the difference q - T between the body temperature and the temperature T of its surrounding medium. Set up the differential equation, solve it to find the general solution for q(t), and use it to find the time of death in this case.
•  
• Determine the values of k for which x^k is a solution of the 2nd order d.e.
• x²y'' - 3xy' + 3y = 0 , (x > 0)

Sample Problems for Test #2

• Show that the general solution to y'' - 2y = 0 may be written as y = c₁ sinh (sqrt(2) x) + c₂ cosh(sqrt(2) x) without solving the associated characteristic equation.
•  
• Solve the following linear homogeneous equations.
  • i) 2y'' + 6y' - y = 0
  • ii) y'' + 2y' + 5y = 0
  • iii) y'' - 18y' + 81y = 0
•  
• Given that the equation y'' + x^-1 y' - x^-2 y = 0 has the general solution y = c₁ x + c₂ x^-1 , find the general solution to y'' + x^-1 y' - x^-2 y = 4x^-1 using variation of parameters.
•  
• Form a trial solution to use if you were going to solve y'' - 3y' = x² e^3x - sin x + 8 by the method of undetermined coefficients, but do not actually find the general solution.
•  
• A mass of 50 gm is attached to a steel spring, stretching it an additional 5 cm from its natural length. If the mass is started in motion with a velocity if 20 cm/sec in the downward direction and there is no air resistance, the motion of the mass is an undamped free vibration. What function u(t ) describes its displacement from equilibrium at time t and what are the associated natural (circular) frequency, amplitude, and phase angle?
•  
• Given that y₁(x) = x is one solution of x²y'' + 2xy' - 2y = 0, determine a second solution.
• Solve: y'' - 2y' + y = e^x - 3x + 1

Sample Problems for Test #3

• Use the definition of the Laplace transform to find L{f(t)} where f(t) = 1 for 0 <= t < 1, f(t) = -1 for 1 <= t < 2, f(t) = 0 elsewhere.
• Find the Laplace transform of the following functions.
  • i) 3t² + 2 cos 2t ii) e^-3t sinh 4t iii) sin²t
• Find the inverse Laplace transform of the following functions.
• i) 4s⁶ ii) 2s - 3s² - 4s + 8 iii) (2(s - 1)e^-2s ) / (s² - 2s + 2)
•  
• Solve this initial value problem using the Laplace transform.
• y" - 9y' + 18y = 54 y(0) = 0, y'(0) = -3
•  
• Use the convolution theorem to solve for f(t) in the following integral equation.
• f(t) = t² - 2 Int(0,t)[ f(t - x) sinh 2x dx]
•  
• If A is a Hermitian matrix, is it necessary to find a generalized eigenvector when solving the system x' = Ax ? Why or why not?
• Solve the system x' = Ax , where x = [ x₁ , x₂ , x₃ ]^T and A = [ 1 -2 2 ; -2 1 -2 ; 2 -2 1 ] .
• Classify the critical point of a 2x2 linear system as to type and stability if the eigenvalues of the system are:
• i) real, unequal, and both negative.
• ii) equal and positive with two linearly independent eigenvectors.
• iii) purely imaginary.
•  
• Solve the following linear system, classify the critical point (0, 0) as to type and stability and sketch several trajectories in the phase plane. dx/dt = Ax where A = [ 2 -1 ; 3 -2 ].

Sample Abstracts of Projects from Fall, 2000

• Modeling the Spread of an Infectious Disease by Rachael Bryk
• Chemical Kinetics by Mondar Ahmed
• Lorenz Equations by Angel Beardslee
• Rate of Memorization Model by Susan Carlo

Past Project Titles

• The Solow Model for Economic Growth (Katie Fitzpatrick)
• Linear and Non-linear Falling Bodies (Mondar Ahmed, Jeff Cox)
• Mass-Spring Systems (Andrea Bourey)
• Modeling of Heat Transfer in a 3-Compartment House (Cara Huldie, Maura Regan)
• Modeling of the Diffusion of Lead in the Human Body (Jennifer Snyder)
• Biomolecular Chemical Reactions (Tim Burke)
• The Celestial Mechanics of Lagrange
• The Hanging Cable Problem
• Heat Transfer through a Solid Rod
• The Brachistochrone Problem
• The Tautochrone Problem
• History of the Wronskian
• Bessel Functions and Kepler's Equation
• The Bernoulli Immunity Differential Equation for the Spread of Smallpox
• The Deflected Beam problem
• The Oscillating Pendulum problem
• Malthusian Law of Growth
• Schrodinger's Equation
• Derivation of the Wave Equation
• Theorem on Separability
• Predator-Prey Systems of ODE's
• History and Motivation of Euler's Equation: Surface area of Minimum Resistance through a Medium.
• Chaos via the Lorenz equations

• Modeling the Spread of an Infectious Disease by Rachel Brik

Modeling physical, biological, and social phenomena using differential equations is a powerful tool for providing attainable answers to obscure questions. In this case, a non-linear system of first-order equations is used to model the way a disease spreads through a population.

Here x(t) represents the number of people who are susceptible to the disease and y(t) represents the number of people who are currently infected with the disease. The group of susceptibles is reduced by one person each time one of them becomes infected. One relevant question is, "When does infection occur?" Infection can occur during the times when susceptibles are in contact with infecteds. Therefore, the rate of change dx/dt of the number of susceptibles is proportional to the product of x(t) and y(t). The first equation above reflects this where a > 0 is a constant that measures how fast the disease is transmitted. The second equation is derived from the fact that the rate of change dy/dt of infecteds is likewise positively affected by the product of x(t) and y(t) but negatively affected by the current population y(t). In other words, the rate slows as y(t) increases.

This system was solved numerically by Maple 6 using feasible values for a and b and initial conditions provided by examining AIDS data for the state of Pennsylvania from 1990 to the present. The vector field and trajectory are shown below.

The graphs for x and y versus t were then compared to the actual data. A reasonable match was then found by modifying the values of a and b.

• Chemical Kinetics by Mondar Ahmed

The rate of a chemical reaction is the change in concentration of a substance per unit time. During a reaction the concentration of reactants decrease and the products increase. Different expermental techniques are used to guage the rate of a reaction such as colour, pH, and electric conductivity. Readings are taken at definite time intervals which allows the prediction of the rate at any concentration. The relation between the rate and the concentration can be mathematically set up as a system of first-order differential equations. For a simple reaction x + y --> z, the representation could be:

where x(t), y(t) are the amounts of the involved substances at time t. The parameters m and n are the orders of the reaction with respect to x and y respectively. Such systems are useful because they allow us to predict how a reaction will proceed and, in particular, can assist in controlling the reactions in organic chemistry where every step and rate must be precisely controlled in order to get desired products. I used Maple 6 to find numerical solutions to this system for various initial conditions and reaction orders. It is interesting to determine when the substances achieve equilibrium depending on whether x(0) > y(0) or x(0) < y(0).

• Lorenz Equations by Angel Beardslee

 The behavior of a physical system like the weather on Earth is extremely complicated and so long-range forecasts have had limited success.In 1963 Edward Lorenz was a meterologist who devised a simplified three-dimensional system of first-order equations to mathematically model the weather in order to make predictions.

This simple system led to the field of Chaos Theory. Chaos deals with the largely unpredictable evolution occurring in a deterministic, nonlinear, dynamical system because of sensitivity to initial conditions. This paper finds the equilibrium points to the system for specific values of the parameters s, r, and b. It also uses the Euler method to find numerical solutions for several sets of slightly different initial conditions. Graphs of the x, y, and z functions are given to demonstrate the unpredictablility of the long-term behavior.

• Rate of Memorization Model by Susan Carlo

Human learning, or more specifically the rate of memorization, is a complicated process. This paper explores the model presented in Differential Equations by Blanchard, Devaney, and Hall on p. 133. It is based on the assumption that the rate of learning is proportional to the amount left to be learned. We let L(t) be the fraction of a list of numbers already committed to memory at time t. So L = 0 corresponds to knowing none of the list and L = 1 corresponds to knowing the entire list. The differential equation is

The parameter k depends on a particular individual's rate of learning. It can be determined through experimentation and so I chose to study my personal rate of memorization to determine my k-value. I began by studying a random list of 20 numbers, each consisting of 3 digits, for one minute. I then attempted to recreate as much of the list as possible from my memory in the correct order. I then spent another minute studying the same list of numbers and quizzed myself again. This process continued until I learned the entire list. It wasn't until after learning the entire list that I went back and graded my "quizzes". I then repeated the process for two more lists. I then applied exponential regression techniques to graphs of L versus t to find a function for L(t) and compared it to the solution for the above differential equation. Finally, I modified the above equation to

and compared the regression fits to the solution to this equation. These seemed to be a better match. I think much more analysis could be done to also account for other factors that affect the learning process.

Useful Websites

http://www.mapleapps.com/index.asp (The Maple Applications Center)

http://math.bu.edu/odes (Boston University D.E. projects)

http://diffeq.brookscole.com (Brooks/Cole D.E. Resource Site)

http://www.maa.org (Mathematical Association of America)

http://math.duke.edu/education/ccp/materials/diffcalc/sir/sir3.html(D.E. model for the spread of disease)

Useful Articles from the American Mathematical Monthly

• "How to solve the system x' = Ax." (Vol. 92, No. 5)
• "When is an ordinary differential equation separable?" (Vol. 92, No. 6)
• "Addition theorem in ordinary differential equations" (Vol. 94, No. 9)
• "Coupled linear DE with real coefficients" (Vol. 96, No. 8)
• "The radius of power series solutions to linear d.e." (Vol. 96, No. 9)
• "Pursuing analogies between DE and difference equations" (Vol. 96, No. 9)
• "Why the method of undetermined coefficents works" (Vol. 98, No. 8)
• "Chaotic Motion of a Pendulum with Oscillatory Forcing" (Vol. 100, No.6)
• "Exploring the Brachistochrone Problem" (Vol. 102, No. 4)
• "Aristotle's Fall" (Vol. 105, No. 6)
• "The Velocity Dependence of Aerodynamic Drag" (Vol. 106, No. 2)
• "A Fractal Example in Ordinary Differential Equations" (Vol. 107, No. 2)

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Comments to Dr. Craig M. Johnson, Chair, Dept. of Mathematics: johnsonc@ac.marywood.edu

Last update: January 29, 2001

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